Using Trivial Mathematics to Estimate Prime Counts with Upper and Lower Bounds, and demonstrating why Littlewood's Theorem Needs Reconsidering

Iain Preston
2021 figshare.com  
Although mathematicians have not yet found the underlying pattern to readily determine where each individual prime sits in the number line, much work has focused on the prime count density. In contrast to observable results, however there exists a theorem by Littlewood which states that the prime count π(X) can exceed the Li(X). This paper uses elementary mathematics to firstly provide a formula that generates estimates for π(X) which are less than Li(X), greater than X/ln X, and which always
more » ... ll within a narrow band of an asymptotic formula. Secondly, the paper then uses "rates of change" data to explain all the relationships that grow with X, and making clear why the LT is not a suitable or consistent model for any projections. Accordingly, it is the author's belief the LT should be reconsidered in light of the evidence so as to compute oscillations without exceeding Li(X).
doi:10.6084/m9.figshare.17125640.v1 fatcat:xrcrqrelafggdknonxtftuhbha